Neutron Stars: Formation and Structure

26 Pages·0.18 MB·English

by  Marek Kutschera

NEUTRON STARS: FORMATION AND STRUCTURE∗)
Marek Kutschera
H. Niewodniczan´ski Institute of Nuclear Physics
ul. Radzikowskiego 152, 31-342 Krak´ow, Poland
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9 Institute of Physics, Jagellonian University
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ul. Reymonta 4, 30-059 Krak´ow, Poland
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Abstract:
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A short introduction is given to astrophysics of neutron stars and to physics of dense
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matter in neutron stars. Observed properties of astrophysical objects containing neutron
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stars are discussed. Current scenarios regarding formation and evolution of neutron stars
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in those objects are presented. Physical principles governing the internal structure of
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/ neutron stars are considered with special emphasis on the possible spin ordering in the
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neutron star matter.
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1. INTRODUCTION
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The existence of compact stars stabilized by the neutron pressure was predicted soon
after discovery of the neutron [1]. The idea of neutron stars is thus more than 60 years
old. The first observation was made only 30 years later in 1967. The history of the idea
and of actual discovery of neutron stars can be found in Ref.[1].
The first radio pulsar was discovered by Bell and Hewish in November 1967 [1]. Soon
after the discovery radio pulsars were identified with rotating magnetized neutron stars [1].
Since then a number of astrophysical objects involving neutron stars have been observed
both in radio and X-ray bands. These include millisecond pulsars, binary radio pulsars,
∗) Lectures given at the XXV Mazurian Lakes School of Physics, Piaski, August 1997.
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high-mass X-ray binaries and low-mass X-ray binaries. Recently, an isolated nonpulsating
neutron star which radiates thermally was observed by the Hubble Space Telescope [2].
The star is less than 130 pc from Earth.
The aim of these lectures is to give an introduction to both astrophysics of neutron
stars and physics of neutron star structure. In Sect.2 a brief review of the astrophysical
objects which contain neutron stars is given. I will shortly describe observed properties of
these objects. In Sect.3 current scenarios of formation of neutron stars in these objects are
presented. In Sect.4 I will outline some basic ideas regarding the neutron star structure.
An emphasis will be given to the behaviour of proton component of the neutron star
matter. In particular, a possible spin ordering in the neutron star core and its implications
for neutron stars are discussed.
2. ASTROPHYSICS OF NEUTRON STARS
Astrophysicalobjectsinhydrostaticequilibriumfallinfourdifferent categoriesaccord-
ing to the physical nature of pressure which supports them against gravity. The source
of pressure in normal stars and in white dwarfs is the kinetic energy of, respectively, a
thermal plasma and a degenerate electron gas. In planets and neutron stars the pressure is
due to the interaction energy of, respectively, a condensed atomic matter and a condensed
nucleon (hadronic) matter.
Here we are interested mostly in neutron stars. From the fact that hadronic pressure
becomes relevant only at a sufficiently high density when the nucleons start to feel their
mutual repulsion we can infer the density and the size of neutron stars. The repulsive
core of nucleon-nucleon interactions start to dominate at densities exceeding the saturation
density ofnuclear matter, n > n = 0.17fm−3 (ormassdensity ρ > ρ = n (m −B/c2) ≈
0 0 0 P
2.7×1014g/cm3). The radius of a 1M star of density ρ is R ≈ 12km. Hence neutron
⊙ 0
stars, objects stabilized by the nucleon (hadron) pressure, have densities of the order of
nuclear matter density and radii of the order of 10km.
Neutron stars, discovered initially as radio pulsars, are responsible for a number of
astrophysical phenomena. Below we discuss in somedetail astrophysical objects containing
neutron stars.
2.1. Radio pulsars
Radio pulsars are rapidly rotating strongly magnetized neutron stars. Their rotation
period, P, is in the range 0.033s ≤ P ≤ 4.0s. Pulsars emit radiation at the expense of
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the rotational energy. As a result, they spin down. The spin-down rate P is measured to
be ∼ 10−12 −10−16ss−1. Measurements of the period, P, and its derivative, P˙, allow us
to obtain the magnetic field of the pulsar, B, assuming that the spin-down is due to the
dipole magnetic braking. A simple formula for the pulsar ”luminosity” is
2R6 2π
L = ( )4B2 = −E˙ , (1)
pulsar 3c3 P rot
where the pulsar rotational energy is E = I(2π/P)2/2. From Eq.(1) we find that the
rot
magnetic field is B = 3.2×1019pPP˙, with P in seconds. In this estimate, typical values
of the moment of inertia, I = 1045gcm2, and the radius, R = 10km, are used. Pulsar
magnetic fields are found to be B ∼ 1011 −1013G.
The total radio luminosity of pulsars is generally ∼ 1026−1030erg/s. These values are
some two to six orders of magnitude below the rate E˙ of rotational energy dissipation.
rot
Pulsars obviously emit energy at other wavelengths.
One of the best studied pulsars is the Crab pulsar, PSR 0531+21, with the period
P = 0.033s. The neutron star (Crab pulsar) and the Crab nebula in which the pulsar is
embedded are remnants of the supernova in the year 1054. The Crab pulsar is thus about
940 years old. It is the youngest known pulsar. The Crab pulsar is observed optically
as a faint star near the center of the Crab nebula. The pulsar radiates also in X- and
γ-rays. In all bands, at radio frequences, in visible light, in X-rays and in γ-rays the signal
displays the same pulse profile, Fig.1. According to current theories of pulsar emission,
the pulsating radiation at all wavelengths is produced by the same magnetospheric activity
which generates the radio signal. This means that the visible light is not emitted from
the surface of the neutron star but rather from an extended magnetosphere. Optical
observations of the Crab pulsar do not allow us to constrain the radius of the star.
2.1.1. Millisecond pulsars
There exists a clearly distinct category of radio pulsars, millisecond pulsars, with
short periods in the range 1.56ms ≤ P ≤ 100ms and very small period derivatives, P˙ ∼
10−17 − 10−21. The fact that corresponding magnetic fields are much weaker than for
ordinary pulsars, B ∼ 108 − 1010G, strongly suggests that millisecond pulsars should be
treated as a different class of radio pulsars. Almost one half of all millisecond pulsars
are members of binary systems, which is another distinct feature of these objects. The
fastest known pulsar, PSR 1937+21, has the period P = 1.56ms and P˙ = 10−19. Some
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researchers [3] postulate that millisecond pulsars with lowest magnetic fields, B ∼ 108G,
are of different origin than those with higher fields.
2.2. X-ray binaries
X-ray observations made by satelliteexperiments have revealed theexistence ofbinary
systems composed of the neutron star with the companion which is a normal star. There
are two classes of such binary stars, high mass X-ray binaries and low mass X-ray binaries.
In both cases, the neutron star is accreting matter from the companion, which results in
X-ray emission. However the pattern of the X-ray emission is different for both classes.
2.2.1. High mass X-ray binaries
The X-ray flux of high-mass X-ray binaries displays a periodic modulation, hence
these objects are also called X-ray pulsars. The companion of the neutron star is usually
a massive O or B star with the mass ∼ 10M − 40M . The neutron star is strongly
⊙ ⊙
magnetized with the magnetic field typical for normal radio pulsars, B ∼ 1012 − 1013G.
Matter accreted onto the neutron star moves along the magnetic field lines and hits the
neutron star surface in the vicinity of the magnetic pole. In this way, a hot spot emitting
X-rays is formed around the magnetic pole. Rotation of the neutron star around an axis
inclined with respect to the magnetic axis produces the pulsar effect. The rotation period
of X-ray pulsars ranges from P ∼ 0.1s to a fraction of an hour. The slowest X-ray pulsar
has the period P ∼ 1400s [4]. Since heavy stars live short the age of high mass X-ray
binaries is estimated to be less than 107y. Recent review of X-ray pulsars is given in
Ref.[5].
One should note that for some accreting X-ray pulsars the rotation period decreases,
P˙ < 0. Apparently, these pulsars are spinning up due to accretion of matter.
2.2.2. Low mass X-ray binaries
The companion of the neutron star in low mass X-ray binaries is a faint star with
the mass M ≤ 1.2M . The X-ray emission displays bursts which repeat in several hours.
⊙
Low mass X-ray binaries are also named X-ray bursters. The X-ray bursts are thought
to be produced by a thermonuclear flash of the accreted nuclear fuel. Accreted matter
spreads over the surface of the neutron star. This occurs because the magnetic field is
low, B ≤ 1010G. Continuous accretion from the companion increases the density and
temperature at the base of the fuel shell. When the condition of instability is reached, a
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nuclear flash occurs. A new shell of fuel is then accreted and the cycle repeats. The X-ray
busters are rather old objects, with the age more than 108 years.
2.3. Binary radio pulsars
An important class of neutron star binaries are systems where the neutron star is
detected as a radio pulsar. Such systems are binaries with no mass transfer. The best
studiedbinaryradiopulsaristheHulseandTaylorpulsar, PSR1913+16[6]. Generally, one
can distinguish two subclasses of these objects, close systems with a compact companion,
PSR 1913+16 subclass, and loose systems with the companion being low-mass faint star,
PSR 1953+29 subclass. In the former case the orbit is tight and eccentric with the orbital
period P ∼ 1day. The compact companion is a neutron star or a massive white dwarf
orb
(M ≈ 1.4M ). In the latter case the system is more loose with the orbital period P ∼
⊙ orb
1year and with circular orbit. The companion is a low mass helium white dwarf with mass
M ≤ 0.4M . At present six double neutron stars are known.
⊙
Observations of double neutron stars are very important. Shrinking of the orbit in
the pulsar PSR 1913+16 is the best indication available of gravitational radiation. Also,
precise determination of masses of both neutron stars is possible for double neutron stars.
This is crucial for constraining models of neutron star interior.
Binary radio pulsars play also important role in our understanding of the evolution of
neutron stars in binaries, as is discussed below.
2.4. Thermally radiating single neutron stars
Astrophysicists expect that there are some ∼ 108 neutron stars in our Galaxy. These
are remnants of previous generations of massive stars which produced all the metals (i.e.
Z > 2 elements) observed at present. Many of these stars should be hot enough and
close enough to be observable as X-ray sources. Recently, the first such star was observed
both in X-ray band [7] and in UV and visible band by the Hubble Space Telescope [2].
The neutron star is hot and it was first observed as a strong X-ray source by the ROSAT
satellite. The star emits thermal radiation and is located about 130 pc from Earth. This
allows for the first time to directly constrain the radius of a neutron star to be R ≤ 14km
[2].
2.4.1. Invisible neutron stars
Nonrotating and nonaccreting neutron stars which have cooled down are virtually un-
detectable by conventional means. Such stars contribute to the dark matter of the Galaxy
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in the form of massive compact objects. Recently, gravitational microlensing experiments
have detected a population of such objects in the galactic halo [8]. Future microlensing
observations are crucial for determination of the contribution of invisible massive baryonic
objects to the total mass of the Galaxy. In particular they should answer the question if
there is a halo of neutron stars around our Galaxy. Such a halo composed of stars which
obtained sufficiently high initial velocity at birth would explain scarcity of local thermally
radiating neutron stars.
2.5. Physical parameters of neutron stars
Observations of neutron stars in all astrophysical objects discussed above allow us
to determine some of their physical parameters. The relevant parameters are the mass,
the radius, the temperature, the age and the magnetic field. Generally, simultaneous
measurement of all above parameters for a given neutron star is not possible.
2.5.1. Neutron star masses
Presently, masses of about 20 neutron stars in binary systems are determined. Among
them are six double neutron star binaries, PSR B1913+16 [9], PSR B1534+12 [10], PSR
B2303+46 [11], PSR B2127+11C [12], PSR B1820-11 [13] and PSR J1518+4904 [14]. For
three of them, PSR B1913+16, PSR B1534+12 and PSR B2127+11C, precise measure-
ments of mass of both neutron stars are now available. The masses are found to be
M = 1.44M and M = 1.39M (PSR B1913+16), M = 1.34M and M = 1.34M
1 ⊙ 2 ⊙ 1 ⊙ 2 ⊙
(PSR B1534+12) and M = 1.35M and M = 1.36M (PSR B2127+11C). All masses
1 ⊙ 2 ⊙
lie in a rather narrow interval, 1.3M < M < 1.5M . Masses of neutron stars in X-ray
⊙ ⊙
pulsars are also consistent with these values although are measured less accurately. The
measured masses of neutron stars apparently do not exceed the maximum mass which is
about 1.5M . We shall discuss possible implications of this upper limit in the last section.
⊙
2.5.2. Neutron star radii
The first observation of thermally radiating nonpulsating neutron star RX J185635-
3754 [2,7] constrains the radius of the star to be R < 14km. Generally, radii of neutron
stars are not directly observable. One can infer, however, some plausible values from model
calculations of X-ray bursters which are of the order of 10km.
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2.5.3. Surface temperature of neutron stars
The X-ray and optical spectrum of the neutron star RX J185635-3754 corresponds to
the black body with k T ≈ 60eV. Also, the X-ray flux from about 14 pulsars has been
B
detected [15]. The spectrum of photons is more difficult to obtain. If measured, it is often
not consistent withthe thermal emission but is rather dominated by a hard component due
to the magnetospheric activity. Only for four pulsars, PSR 0833-45 (Vela), PSR 0630+18
(Geminga), PSR 0656+14 and PSR 1055-52, softer blackbody component corresponding
to surface thermal emission is determined.
2.5.4. The age of pulsars
Pulsar ages are estimated by measuring their spin-down rates. Pulsars spin down due
to conversion of rotational energy into radiation. A simple spin-down relation is assumed,
ν˙ = Kνn, where ν is the rotation frequency, and n is the braking index. The constant
K for magnetic braking is proportional to d2/I, where d is the magnetic moment of the
star and I is the moment of inertia. If the energy loss is through radiation from a dipolar
magnetic field, the braking index is n = 3. The spin-down age of pulsar is then τ = −ν/2ν˙.
Thespin-down agewithn = 3iscommonlyusedforpulsars. Itsapplicability,however,
is questioned by recent measurement of the braking index of the Vela pulsar [16], which
gives n = 1.4 ± 0.2. This value implies that previous estimate of the age of Vela pulsar
should increase by a factor ∼ 3.
Typical values of the spin-down age for radio pulsars are 107y. For millisecond pulsars
the spin-down age exceeds 109y.
2.5.5. Magnetic fields of neutron stars
Magnetic fields of radio pulsars are inferred from the spin-down relation assuming
dipolar magnetic field. A striking feature is the bimodal distribution of pulsar magnetic
fields. Usual pulsars have strong magnetic field, B ∼ 1011 − 1013G, whereas millisecond
pulsars possess much weaker fields, B ∼ 108−1010G. For some X-ray pulsars the magnetic
field is measured directly, by observation of absorption features interpreted as cyclotron
lines [17]. The values are in the range found for normal pulsars. It should be noted that
neutron stars in the X-ray bursters have, if any, weaker fields, B < 1010G.
The magnetic field of neutron stars could also serve as a probe of the neutron star
equationofstate(EOS)ifitspresence isdetermined by thepropertiesofdense matter. The
bimodal distribution of pulsar magnetic fields strongly suggests existence of a magnetic
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phase transition in the neutron star matter [17]. There is a possibility, however, that some
component of the magnetic field of a neutron star is inherited from the progenitor.
The magnetic phase transition in the neutron star matter is discussed in Sect.4.
3. FORMATION OF NEUTRON STARS
Neutron stars are the final product of stellar evolution. The evolution of single stars
is significantly different than the evolution of stars in binaries. The formation scenario of
neutron stars in binary systems reflects this fact. We briefly discuss below both cases.
3.1. Evolution of single stars
Life of single stars with the mass M ≥ 8M is terminated by the supernova explosion.
⊙
The explosion is a result of the gravitational collapse of the star’s iron core whose mass
has exceeded the Chandrasekhar mass, M ≈ 1.5M , the maximum mass which can
Ch ⊙
be supported by pressure of the electron gas. The collapse is halted when the density
increases sufficiently for the nucleons to feel the repulsive core of their mutual interaction.
The nuclear interactions produce very steep pressure gradient which is able to stop the
collapse. Rapid release of the gravitational energy, which is carried mostly by neutrinos,
causes the violent explosion of outer layers of the star. The total amount of the energy
released in the gravitational collapse of the stellar core is ∼ 1053erg. Only ∼ 1% of
this energy is needed to expel the mantle and envelope of the collapsing star. The basic
scenario of the Type II supernova explosion was spectacularly confirmed by detection by
theKamiokandeexperimentoftheneutrinoburst associatedwiththesupernova SN1987A.
The energy of neutrinos and the distance to the Large Magellanic Cloud, where SN 1987A
occurred, allows to estimate the total energy carried by neutrinos to be ∼ 1053erg.
3.2. Evolution of binaries
Evolution of stars in close binaries in which stars can exchange mass proceeds differ-
entlythanforsinglestars. Evolutionofbinariesrelevanttoneutron starformationdepends
on the initial mass and separation of the primary and secondary star. Below we discuss
the origin of high mass X-ray binaries and low mass X-ray binaries and their evolutionary
links to binary and millisecond pulsars.
3.2.1. Formation of high mass X-ray binaries
The progenitor of the high mass X-ray binary is a binary system with the primary star
heavier than the O or B star present in such X-ray binaries, M > M . This star evolves
1 2
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quickly and explodes as a supernova. In general, the binary system can be disrupted by
the supernova explosion. However, if the mass ejected from the system is less than the
mass left,
M < M +M (2)
ejected 2 NS
where M is the mass of the neutron star, the binary binding can survive [18]. Also, a
NS
suitable natal kick imparted to the neutron star at birth in asymmetric supernova explo-
sion, can save the binary from disruption. For the produced system to become an X-ray
binary sufficient accretion onto the neutron star is required. Since the newly formed neu-
tron star has a strong magnetic field, it becomes an X-ray pulsar when it accretes matter
from the normal companion. Such an interacting binary system can be formed for a range
of masses and orbital separations of the primary and secondary stars of the progenitor.
3.2.2. Origin of double neutron stars
There exist scenarios linking the formation of the binary pulsars of PSR 1913+16
class to the evolution of high mass X-ray binaries. In these models the evolution proceeds
through the common envelope phase. The companion of the neutron star in the high mass
X-ray binary, which is primary as far as the mass is concerned, evolves quite fast. When it
approaches the red giant phase, its expanding hydrogen envelope can engulf the neutron
star. The binary system enters a hypothetical period of common envelope evolution. The
neutron star is expected to spiral inward due to friction, getting closer to the helium core.
In this process, the envelope which is weakly bound is thought to become expelled. As a
result, a binary system of the neutron star and the helium star, is formed. The mass of
the helium star is about 3M . Such a helium star after some time undergoes core collapse
⊙
and explodes as a supernova which leaves a remnant neutron star. Detailed considerations
suggest[19]thatonlyinanasymmetricsupernova explosionabinarysystemoftwoneutron
stars can be formed.
The binary pulsar PSR 1913+16, which is a double neutron star, has likely followed
such an evolutionary track. Similar evolutionary considerations are valid for other pairs
of neutron stars. These systems are observed as binary radio pulsars with a neutron
star companion. An important question is which of the two neutron stars is the radio
pulsar. The fact, that the pulsar in PSR 1913+16 has period 59ms and low magnetic field
B ∼ 2× 1010G suggests that it is a recycled pulsar with the age ∼ 108y. Newly formed
neutron star is expected to have much stronger magnetic field, B ∼ 1012G, and to live as
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a radio pulsar only ∼ 107 years. Hence it is more likely that we observe as a radio pulsar
the older neutron star in this binary.
The orbits of binary pulsars with neutron star companions shrink due to gravitational
radiation. The two neutron stars will collide/merge in some 109y. The collision/merger is
unavoidable. In such a violent event a lot of energy will be liberated in a rather short time
interval and the collision can manifest itself as a powerful explosion. Many researchers
consider such a neutron star collision the best candidate for gamma-ray bursters.
3.2.3. Formation of low-mass X-ray binaries
The origin of low mass X-ray binaries is more difficult to explain [20]. Most of evo-
lutionary models base on common envelope evolution prior to the supernova explosion.
The major problem is the asymmetry of masses of assumed progenitors of these binaries.
The masses of donor stars, which are companions of neutron stars, are low, M ≤ 1.2M ,
⊙
and these stars are unevolved. The mass of the primary of the progenitor must be high
enough in order to produce a neutron star. It can exceed the secondary mass, which is the
donor mass of low mass X-ray binary by a factor ≥ 20. The evolutionary sequence leading
to the low mass X-ray binary consists of several steps. First, a common envelope forms,
due to small time scale of nuclear evolution of the massive primary. During this phase the
hydrogen envelope is dispersed and the binary orbit is shrunk. Otherwise, if the primary
explodes as a supernova before the envelope is expelled the system becomes disrupted.
After the common envelope phase the binary system with a helium star emerges. The
chance the system survives the explosion of the helium star is increased, in particular the
asymmetric explosion imparting a proper kick to the neutron star can help.
An alternative scenario of formation of low mass X-ray binaries is the accretion-
induced collapse of the white dwarf. In this model, the Chandrasekhar mass white dwarf
collapses to the neutron star when some matter is accreted onto it. The progenitor binary
consists of two low mass stars, with the heavier one becoming the carbon white dwarf.
It remains an open question if the accretion induced collapse of the white dwarf occurs
in nature. One should remember that the carbon white dwarf contains a lot of nuclear
fuel which can undergo thermonuclear flash, that completely disrupts the star. Such an
explosive carbon burning of a white dwarf in a binary is a successful model of the Type I
supernovae. No remnant star is left in such an explosion.
Those binaries which sustain low accretion rates for a very long time are observed
as low mass X-ray binaries. Magnetic fields of neutron stars in these binaries seem to be
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